#include <eim.hpp>

Detailed Description

template<typename ModelType>
class Feel::EIM< ModelType >

Empirical interpolation of a function to obtain an affine decomposition.

We are given a function $g (\: \cdot\: ; \mu) \in L^{\infty} (\Omega)$ of sufficient regularity.

Offline Stage

To begin, we choose $\mu^g_1$ , and define $S^g_1 = \{ \mu^g_1 \}$ , $\xi_1 \equiv g (x ; \mu^g_1)$ , and $W^g_1 = {\rm span} \: \{\xi_1 \}$ ; we assume that $\xi_1 \neq 0$ . Then, for $M \geq 2$ , we set $\mu^g_M = \arg \max_{\mu \in \Xi^{^g}}\inf _{z \in W^g_{M-1}} \|g (\: \cdot \: ; \mu) - z \|_{L^{\infty} (\Omega)}$ , where $\Xi^g$ is a suitably fine parameter sample over ${\mathcal{D}}$ . We then set $S^g_M = S^g_{M-1} \cup \mu^g_M$ , $\xi_M = g (x;\mu^g_M)$ , and $W^g_M = {\rm span} \: \{ \xi_m, \: 1 \leq m \leq M \}$ . Note that, thanks to our truth approximation, $\mu^g_M$ is the solution of a standard linear program.

We suppose that $M_{\max}$ is chosen such that the dimension of $\{g (\: \cdot \: ; \mu) \: | \: \mu \in \mathcal{D}\}$ exceeds $M_{\max}$ .

We now construct nested sets of interpolation points $T_M = \{ t_1, \ldots, t_M \}$ , $1 \leq M \leq M_{\max}$ . We first set $t_1 = \arg \: {\rm ess} \: \sup_{x \in \Omega} | \xi_1 (x)|$ , $q_1 = \xi_1 (x) / \xi_1 (t_1)$ , $B^1_{11} = 1$ . Then for $M = 2, \ldots, M_{\max}$ , we solve the linear system $\sum^{M-1}_{j = 1} \: \sigma^{M-1}_j \: q_j(t_i) = \xi_M (t_i)$ , $1 \leq i \leq M-1$ , and set $r_M (x) = \xi_M (x) - \sum^{M-1}_{j = 1}\: \sigma^{M-1}_j \: q_j (x)$ , $t_M = \arg \: {\rm ess} \: \sup_{x \in \Omega} |r_M (x)|$ , $q_M (x) = r_M (x) /r_M (t_M) $ , and $B^M_{i \: j} = q_j (t_i)$ , $1 \leq i,j \leq M$ .

Online Stage

Our coefficient function approximation is the interpolant of $g$ over $T_M$ : $g_M (x ; \mu) = \sum^M_{m = 1} \beta_m (\mu) \: q_m (x)$ , where $\sum^M_{j = 1} \: B^M_{i \: j} \: \beta_j (\mu) = g (t_i ; \mu)$ , $1 \leq i \leq M$ . We define $\varepsilon_M (\mu) \equiv \| g (\: \cdot \: ; \mu) - g_M (\: \cdot \: ; \mu)\|_{L^{\infty} (\Omega)}$ .

Author: Christophe Prud'homme chris.nosp@m.toph.nosp@m.e.pru.nosp@m.dhom.nosp@m.me@fe.nosp@m.elpp.nosp@m..org

See Also

Inheritance diagram for Feel::EIM< ModelType >:

Public Types
Typedefs
typedef ModelType	model_type

typedef ModelType::value_type	value_type

typedef Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic >	matrix_type

typedef matrix_type::ColXpr	column_type

typedef Eigen::Matrix< double, Eigen::Dynamic, 1 >	vector_type

typedef ModelType::node_type	node_type

typedef ModelType::functionspace_type	functionspace_type

typedef boost::shared_ptr < functionspace_type >	functionspace_ptrtype

typedef functionspace_type::element_type	element_type

typedef functionspace_type::element_ptrtype	element_ptrtype

typedef ModelType::solution_type	solution_type

typedef ModelType::parameterspace_type	parameterspace_type

typedef ModelType::parameter_type	parameter_type

typedef parameterspace_type::sampling_ptrtype	sampling_ptrtype

typedef boost::tuple< double, Eigen::Matrix< double, nDim, 1 > >	space_residual_type

typedef boost::tuple< double, parameter_type >	parameter_residual_type

Public Member Functions
Constructors, destructor
	EIM ()

	EIM (po::variables_map const &vm, model_type *model, sampling_ptrtype sampling, double __tol=1e-8)

	EIM (EIM const &__bbf)

	~EIM ()

Accessors
size_type	nDOF () const

std::vector< element_type > const &	q () const

element_type const &	q (size_type __m) const

size_type	mMax () const

bool	isOfflineDone () const

Mutators
void	setTrainSet (sampling_ptrtype pset)

Public Member Functions inherited from Feel::CRBDB
	CRBDB ()
	default constructor

	CRBDB (std::string prefixdir, std::string name, std::string dbprefix, po::variables_map const &vm)
	constructor from command line options

	CRBDB (CRBDB const &)
	copy constructor

virtual	~CRBDB ()
	destructor

CRBDB &	operator= (CRBDB const &o)
	copy operator

std::string const &	prefixDirectory () const

std::string const &	name () const

std::string const &	dbFilename () const

fs::path	dbLocalPath () const

fs::path	dbSystemPath () const

fs::path	lookForDB () const

po::variables_map	vm ()

po::variables_map	vm () const

bool	isDBLoaded () const

void	setDBFilename (std::string const &filename)
	set the DB filename

Static Public Attributes
static const uint16_type	nDim = ModelType::nDim

Protected Attributes
matrix_type	M_B

std::vector< element_type >	M_g

std::vector< size_type >	M_index_max

bool	M_is_read

bool	M_is_written

size_type	M_M

model_type *	M_model

std::string	M_name

bool	M_offline_done

std::vector< element_type >	M_q

std::vector< node_type >	M_t

double	M_tol

sampling_ptrtype	M_trainset

po::variables_map	M_vm

size_type	M_WN

size_type	MM_max

Methods
class	boost::serialization::access

void	saveDB ()

bool	loadDB ()

vector_type	beta (parameter_type const &mu) const
	Online stage of the coefficient-function interpolation. More...

vector_type	beta (parameter_type const &mu, solution_type const &T) const

vector_type	beta (parameter_type const &mu, size_type M) const

vector_type	beta (parameter_type const &mu, solution_type const &T, size_type M) const

std::vector< double >	studyConvergence (parameter_type const &mu, solution_type &solution) const

void	computationalTimeStatistics (std::string appname)

element_type	residual (size_type M) const

parameter_residual_type	computeBestFit (sampling_ptrtype trainset, int __M)

element_type	operator() (parameter_type const &mu, int N) const

element_type	operator() (parameter_type const &mu, solution_type const &T, int N) const

void	orthonormalize (std::vector< element_type > &)

template<class Archive >
void	serialize (Archive &__ar, const unsigned int __version)

Additional Inherited Members
Protected Member Functions inherited from Feel::CRBDB
template<class Archive >
void	load (Archive &ar, const unsigned int version)

template<class Archive >
void	save (Archive &ar, const unsigned int version) const

void	setIsLoaded (bool isloaded)

Member Function Documentation

template<typename ModelType>

vector_type Feel::EIM< ModelType >::beta ( parameter_type const & mu ) const

inline

Online stage of the coefficient-function interpolation.

Our coefficient function approximation is the interpolant of $g$ over $T_M$ : $g_M (x ; \mu) = \sum^M_{m = 1} \beta_m (\mu) \: q_m (x)$ , where $\sum^M_{j = 1} \: B^M_{i \: j} \: \beta_j (\mu) = g (t_i ; \mu)$ , $1 \leq i \leq M$ . We define $\varepsilon_M (\mu) \equiv \| g (\: \cdot \: ; \mu) - g_M (\: \cdot \: ; \mu)\|_{L^{\infty} (\Omega)}$ .

Note that $\Omega$ is given and $D^$ is handled by the parameterset_type data structure.

References Feel::EIM< ModelType >::beta().

template<typename ModelType>

bool Feel::EIM< ModelType >::isOfflineDone ( ) const

inline

Check whether the offline stage has been executed and the database created.

Returns: true if the offline has been executed and the DB saved, false otherwise

template<typename ModelType>

bool Feel::EIM< ModelType >::loadDB ( )

inlinevirtual

load the CRB database

Reimplemented from Feel::CRBDB.

References Feel::CRBDB::lookForDB().

template<typename ModelType>

size_type Feel::EIM< ModelType >::nDOF ( ) const

inline

Returns: the number of DOF in space discretization

template<typename ModelType >

void Feel::EIM< ModelType >::orthonormalize ( std::vector< element_type > & __Z )

orthonormalize

References Feel::inner_product().

template<typename ModelType>

std::vector<element_type> const& Feel::EIM< ModelType >::q ( ) const

inline

return the set of reduced basis functions associated with the eim

template<typename ModelType>

element_type const& Feel::EIM< ModelType >::q ( size_type __m ) const

inline

return the m-th reduced basis function associated with the eim

template<typename ModelType>

void Feel::EIM< ModelType >::saveDB ( )

inlinevirtual

save the CRB database

Reimplemented from Feel::CRBDB.

References Feel::CRBDB::dbFilename(), and Feel::CRBDB::dbLocalPath().

The documentation for this class was generated from the following file:

eim.hpp

Detailed Description

template<typename ModelType> class Feel::EIM< ModelType >

Offline Stage

Online Stage

Public Types

Public Member Functions

Static Public Attributes

Protected Attributes

Methods

Additional Inherited Members

Member Function Documentation

template<typename ModelType>
class Feel::EIM< ModelType >